On the regularity and symmetry of periodic traveling solutions to weakly dispersive equations with cubic nonlinearity

We consider the a priori regularity and symmetry of periodic traveling wave solutions to a class of weakly dispersive equations with cubic nonlinearity. We prove that any continuous periodic traveling wave solution of small amplitude is symmetric if it satisfies the reflection criterion proposed recently by Bruell and Pei. Moreover, we confirm that continuous periodic traveling waves with singular crests or troughs, which correspond to large-amplitude waves, are also symmetric if some cancellation structure appears near the singular points or if the wave profile has only a single singular point per period. Included in the results as particular cases are periodic traveling waves with peak, cusp, or stump type singular structures. The argument and results also apply to dispersive equations with more general nonlinearity which is non-smooth.